My fujifilm finepix s5600 seems to have broken. Anyone have any experience of how much this kind of thing costs to fix?

Problem - it won't take photos (simple, really). It will switch on to photo-viewing mode, and then just won't make the necessary next step into actually switching on to take photos - the screen just goes blank (and yes, I have made sure it isn't on the little electronic view finder). When I try to move it from photo-viewing to photo-taking, you can tell from holding and listening to the camera as well as looking at it that it isn't switching - the normal noise just doesn't happen.

I have an S6500 which is now displaying lines in the EVF like someone drew on it with a white pencil.

That was after about two and a half years ownership. Still takes photos but is obviously starting to go off.

My view is that this sort of equipment is build with the expectation of a three-year lifespan. Some certainly last longer, but it is likely that the manufacturers have carefully calculated when people will want to update to new technology and built to that standard. Same thing with computers. There's an "economic" life and a bit of built-in redundancy to ensure that you get to there.

Yes unfortunately digital equipment (especially compacts) does not seem to be built to last with the way film cameras were. They only have a limited lifetime it looks like yours has reached.

A quick ebay search revealed your camera could be bought used but decent condition for about £20, so I wouldnt bother repairing it (it would come to over this probably), Id just buy either another one, or a new camera.